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			<h1 id="firstHeading" class="firstHeading">Sallen–Key topology</h1>
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				<p>The <b>Sallen–Key topology</b> is an <a href="http://en.wikipedia.org/wiki/Electronic_filter_topology" title="Electronic filter topology">electronic filter topology</a> used to implement second-order <a href="http://en.wikipedia.org/wiki/Active_filter" title="Active filter">active filters</a> that is particularly valued for its simplicity.<sup id="cite_ref-EE315A_Notes_0-0" class="reference"><a href="#cite_note-EE315A_Notes-0"><span>[</span>1<span>]</span></a></sup> It is a <a href="http://en.wikipedia.org/wiki/Degeneracy_%28mathematics%29" title="Degeneracy (mathematics)">degenerate</a> form of a <b>voltage-controlled voltage-source (VCVS) filter topology</b>. A VCVS filter uses a <a href="http://en.wiktionary.org/wiki/super-" class="extiw" title="wiktionary:super-">super</a>-<a href="http://en.wikipedia.org/wiki/1_%28number%29" title="1 (number)">unity</a>-<a href="http://en.wikipedia.org/wiki/Gain" title="Gain">gain</a> voltage amplifier with practically infinite <a href="http://en.wikipedia.org/wiki/Input_impedance" title="Input impedance">input impedance</a> and zero <a href="http://en.wikipedia.org/wiki/Output_impedance" title="Output impedance">output impedance</a> to implement a <a href="http://en.wikipedia.org/wiki/Pole_%28complex_analysis%29" title="Pole (complex analysis)">2-pole</a> (12 dB/octave) <a href="http://en.wikipedia.org/wiki/Low-pass" title="Low-pass" class="mw-redirect">low-pass</a>, <a href="http://en.wikipedia.org/wiki/High-pass" title="High-pass" class="mw-redirect">high-pass</a>, or <a href="http://en.wikipedia.org/wiki/Bandpass" title="Bandpass" class="mw-redirect">bandpass</a> <a href="http://en.wikipedia.org/wiki/Frequency_response" title="Frequency response">response</a>. The super-unity-gain amplifier allows for very high <a href="http://en.wikipedia.org/wiki/Q_factor" title="Q factor">Q factor</a> and <a href="http://en.wikipedia.org/wiki/Passband" title="Passband">passband</a> gain without the use of <a href="http://en.wikipedia.org/wiki/Inductor" title="Inductor">inductors</a>. A Sallen–Key filter is a variation on a VCVS filter that uses a unity-gain amplifier (i.e., a pure <a href="http://en.wikipedia.org/wiki/Buffer_amplifier" title="Buffer amplifier">buffer amplifier</a> with 0 <a href="http://en.wikipedia.org/wiki/Decibel" title="Decibel">dB</a> gain). It was introduced by <a href="http://en.wikipedia.org/w/index.php?title=R.P._Sallen&amp;action=edit&amp;redlink=1" class="new" title="R.P. Sallen (page does not exist)">R.P. Sallen</a> and <a href="http://en.wikipedia.org/w/index.php?title=E._L._Key&amp;action=edit&amp;redlink=1" class="new" title="E. L. Key (page does not exist)">E. L. Key</a> of <a href="http://en.wikipedia.org/wiki/MIT" title="MIT" class="mw-redirect">MIT</a> <a href="http://en.wikipedia.org/wiki/Lincoln_Laboratory" title="Lincoln Laboratory">Lincoln Laboratory</a> in 1955.<sup id="cite_ref-SallenKey_1-0" class="reference"><a href="#cite_note-SallenKey-1"><span>[</span>2<span>]</span></a></sup></p>
<p>Because of its high input impedance and easily selectable gain, an <a href="http://en.wikipedia.org/wiki/Operational_amplifier" title="Operational amplifier">operational amplifier</a> in a conventional <a href="http://en.wikipedia.org/wiki/Operational_amplifier_applications#Non-inverting_amplifier" title="Operational amplifier applications">non-inverting configuration</a> is often used in VCVS implementations.<sup class="Template-Fact" title="This claim needs references to reliable sources from January 2009" style="white-space:nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a></i>]</sup> Implementations of Sallen–Key filters often use an operational amplifier configured as a <a href="http://en.wikipedia.org/wiki/Operational_amplifier_applications#Voltage_follower" title="Operational amplifier applications">voltage follower</a>; however, <a href="http://en.wikipedia.org/wiki/Common_collector" title="Common collector">emitter</a> or <a href="http://en.wikipedia.org/wiki/Common_drain" title="Common drain">source</a> followers are other common choices for the buffer amplifier.</p>
<p>VCVS filters are relatively resilient to component <a href="http://en.wikipedia.org/wiki/Tolerance_%28engineering%29" title="Tolerance (engineering)" class="mw-redirect">tolerance</a>, but obtaining high Q factor may require extreme component value spread or high amplifier gain.<sup id="cite_ref-EE315A_Notes_0-1" class="reference"><a href="#cite_note-EE315A_Notes-0"><span>[</span>1<span>]</span></a></sup> Higher-order filters can be obtained by cascading two or more stages.</p>
<table id="toc" class="toc">
<tbody><tr>
<td>
<div id="toctitle">
<h2>Contents</h2>
 <span class="toctoggle">[<a href="#" class="internal" id="togglelink">hide</a>]</span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Generic_Sallen.E2.80.93Key_topology"><span class="tocnumber">1</span> <span class="toctext">Generic Sallen–Key topology</span></a>
<ul>
<li class="toclevel-2 tocsection-2"><a href="#Interpretation"><span class="tocnumber">1.1</span> <span class="toctext">Interpretation</span></a></li>
<li class="toclevel-2 tocsection-3"><a href="#Example_applications"><span class="tocnumber">1.2</span> <span class="toctext">Example applications</span></a>
<ul>
<li class="toclevel-3 tocsection-4"><a href="#Example:_Low-pass_filter"><span class="tocnumber">1.2.1</span> <span class="toctext">Example: Low-pass filter</span></a></li>
<li class="toclevel-3 tocsection-5"><a href="#Example:_High-pass_filter"><span class="tocnumber">1.2.2</span> <span class="toctext">Example: High-pass filter</span></a></li>
</ul>
</li>
</ul>
</li>
<li class="toclevel-1 tocsection-6"><a href="#VCVS_Example:_Bandpass_configuration"><span class="tocnumber">2</span> <span class="toctext">VCVS Example: Bandpass configuration</span></a></li>
<li class="toclevel-1 tocsection-7"><a href="#See_also"><span class="tocnumber">3</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-8"><a href="#External_links"><span class="tocnumber">4</span> <span class="toctext">External links</span></a></li>
<li class="toclevel-1 tocsection-9"><a href="#References"><span class="tocnumber">5</span> <span class="toctext">References</span></a></li>
</ul>
</td>
</tr>
</tbody></table>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Sallen%E2%80%93Key_topology&amp;action=edit&amp;section=1" title="Edit section: Generic Sallen–Key topology">edit</a>]</span> <span class="mw-headline" id="Generic_Sallen.E2.80.93Key_topology">Generic Sallen–Key topology<span id="Generic_Sallen.E2.80.93Key_topology"></span><span id="Generic_Sallen-Key_topology"></span></span></h2>
<p>The generic unity-gain Sallen–Key filter topology implemented with a unity-gain <a href="http://en.wikipedia.org/wiki/Operational_amplifier" title="Operational amplifier">operational amplifier</a> is shown in Figure&nbsp;1. The following analysis is based on the assumption that the <a href="http://en.wikipedia.org/wiki/Operational_amplifier" title="Operational amplifier">operational amplifier</a> is <a href="http://en.wikipedia.org/wiki/Idealism" title="Idealism">ideal</a>.</p>
<div class="thumb tright">
<div class="thumbinner" style="width:402px;"><a href="http://en.wikipedia.org/wiki/File:Sallen-Key_Generic_Circuit.svg" class="image"><img alt="" src="wikipedia-Sallen%E2%80%93Key_topology_pliki/400px-Sallen-Key_Generic_Circuit.png" class="thumbimage" height="200" width="400"></a>
<div class="thumbcaption">Figure&nbsp;1: The generic Sallen–Key filter topology.</div>
</div>
</div>
<p>Because the operational amplifier (OA) is in a <a href="http://en.wikipedia.org/wiki/Negative_feedback" title="Negative feedback">negative-feedback</a> configuration, its <i>v</i><sub>+</sub> and <i>v</i><sub>-</sub> inputs must match (i.e., <i>v</i><sub>+</sub> = <i>v</i><sub>-</sub>). However, the inverting input <i>v</i><sub>-</sub> is connected directly to the output <i>v</i><sub>out</sub>, and so</p>
<dl>
<dd>
<div style="float:left;"><img class="tex" alt="v_+ = v_- = v_{\text{out}}.\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/cc49a442ced068614e72c2f453ccadfe.png"></div>
<div style="text-align: right;"><img class="tex" alt="(1)\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/09805d958fc7ab3d6c57c02f71afdc39.png"></div>
<br clear="left"></dd>
</dl>
<p>By <a href="http://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws" title="Kirchhoff's circuit laws">Kirchhoff's current law</a> (KCL) applied at the <i>v</i><sub>x</sub> node,</p>
<dl>
<dd>
<div style="float:left;"><img class="tex" alt="\frac{v_{\text{in}}-v_x}{Z_1}=\frac{v_x-v_{\text{out}}}{Z_4}+\frac{v_x-v_-}{Z_2}." src="wikipedia-Sallen%E2%80%93Key_topology_pliki/f97cd201b6b2a3182b9904a9aa951121.png"></div>
<div style="text-align:right;"><img class="tex" alt="(2)\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/ed004ded4d1925b3ebfc6e1ad87a0b0f.png"></div>
<br clear="left"></dd>
</dl>
<p>By combining Equations&nbsp;(1) and&nbsp;(2),</p>
<dl>
<dd><img class="tex" alt="\frac{v_{\text{in}}-v_x}{Z_1}=\frac{v_x-v_{\text{out}}}{Z_4}+\frac{v_x-v_{\text{out}}}{Z_2}" src="wikipedia-Sallen%E2%80%93Key_topology_pliki/dedcecb9421e2cd95cca94de3abd8e5e.png"></dd>
</dl>
<p>Applying Equation&nbsp;(1) and KCL at the OA's non-inverting input <i>v</i><sub>+</sub> gives</p>
<dl>
<dd><img class="tex" alt="\frac{v_x-v_{\text{out}}}{Z_2}=\frac{v_{\text{out}}}{Z_3}," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/b476b6781b76ca5a1312732a0fbd9c77.png"></dd>
</dl>
<p>which means that</p>
<dl>
<dd>
<div style="float:left;"><img class="tex" alt="v_x=v_{\text{out}} \left( \frac{Z_2}{Z_3}+1 \right)." src="wikipedia-Sallen%E2%80%93Key_topology_pliki/4e0719792fe91f42f6f05936214522ab.png"></div>
<div style="text-align:right;"><img class="tex" alt="(3)\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/5c5afe41da9b4223447f2257873c995a.png"></div>
<br clear="left"></dd>
</dl>
<p>Combining Equations&nbsp;(2) and&nbsp;(3) gives</p>
<dl>
<dd>
<div style="float:left;"><img class="tex" alt="\frac{v_{\text{in}}-v_{\text{out}} \left( \frac{Z_2}{Z_3}+1 \right)}{Z_1}=\frac{v_{\text{out}} \left( \frac{Z_2}{Z_3}+1 \right)-v_{\text{out}}}{Z_4}+\frac{v_{\text{out}} \left( \frac{Z_2}{Z_3}+1 \right)-v_{\text{out}}}{Z_2}." src="wikipedia-Sallen%E2%80%93Key_topology_pliki/f61860d7778ae36e9024ab3b368c92eb.png"></div>
<div style="text-align:right;"><img class="tex" alt="(4)\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/f0f23a3efbf8781555d7620e18807b8c.png"></div>
<br clear="left"></dd>
</dl>
<p>Rearranging Equation&nbsp;(4) gives the <a href="http://en.wikipedia.org/wiki/Transfer_function" title="Transfer function">transfer function</a></p>
<dl>
<dd>
<div style="float:left;"><img class="tex" alt="\frac{v_{\text{out}}}{v_{\text{in}}} = \frac{Z_3 Z_4}{Z_1 Z_2 + Z_4(Z_1 + Z_2) + Z_3 Z_4}," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/b35c4e592c2e4486630caaff145e6fe6.png"></div>
<div style="text-align:right;"><img class="tex" alt="(5)\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/c86d078e1c54a8ea06fd98109f739dc9.png"></div>
<br clear="left"></dd>
</dl>
<p>which typically describes a second-order <a href="http://en.wikipedia.org/wiki/LTI_system_theory" title="LTI system theory">LTI system</a>.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Sallen%E2%80%93Key_topology&amp;action=edit&amp;section=2" title="Edit section: Interpretation">edit</a>]</span> <span class="mw-headline" id="Interpretation">Interpretation</span></h3>
<p>If the <img class="tex" alt="Z_4\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/a8acb3fb6e92ea0df0316e407a025170.png"> component were connected to ground, the filter would be a <a href="http://en.wikipedia.org/wiki/Voltage_divider" title="Voltage divider">voltage divider</a> composed of the <img class="tex" alt="Z_1\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/cf1ad6dd33b7b9dc1199fa4b1b38af4d.png"> and <img class="tex" alt="Z_4\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/a8acb3fb6e92ea0df0316e407a025170.png"> components cascaded with another voltage divider composed of the <img class="tex" alt="Z_2\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/ff9fd20614395f85bbd9c61d72631f28.png"> and <img class="tex" alt="Z_3\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/b6de4a12cecd4d0f227024c80513e3c0.png"> components. The buffer <a href="http://en.wikipedia.org/wiki/Bootstrapping_%28electronics%29" title="Bootstrapping (electronics)">bootstraps</a> the "bottom" of the <img class="tex" alt="Z_4\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/a8acb3fb6e92ea0df0316e407a025170.png">
 component to the output of the filter, which will improve upon the 
simple two divider case. This interpretation is the reason why 
Sallen–Key filters are often drawn with the operational amplifier's 
non-inverting input below the inverting input, thus emphasizing the 
similarity between the output and ground.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Sallen%E2%80%93Key_topology&amp;action=edit&amp;section=3" title="Edit section: Example applications">edit</a>]</span> <span class="mw-headline" id="Example_applications">Example applications</span></h3>
<p>By choosing different <a href="http://en.wikipedia.org/wiki/Passivity_%28engineering%29" title="Passivity (engineering)">passive components</a> (<a href="http://en.wikipedia.org/wiki/E.g." title="E.g." class="mw-redirect">e.g.</a>, <a href="http://en.wikipedia.org/wiki/Resistor" title="Resistor">resistors</a> and <a href="http://en.wikipedia.org/wiki/Capacitor" title="Capacitor">capacitors</a>) for <img class="tex" alt="Z_1\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/cf1ad6dd33b7b9dc1199fa4b1b38af4d.png">, <img class="tex" alt="Z_2\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/ff9fd20614395f85bbd9c61d72631f28.png">, <img class="tex" alt="Z_3\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/b6de4a12cecd4d0f227024c80513e3c0.png">, and <img class="tex" alt="Z_4\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/a8acb3fb6e92ea0df0316e407a025170.png">, the filter can be made with <a href="#Example:_Low-pass_filter">low-pass</a>, bandpass, and <a href="#Example:_High-pass_filter">high-pass</a> characteristics. In the examples below, recall that a resistor with <a href="http://en.wikipedia.org/wiki/Electrical_resistance" title="Electrical resistance" class="mw-redirect">resistance</a> <img class="tex" alt="R\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/153fc2a5a0a49d52dda62d96ae0a293f.png"> has <a href="http://en.wikipedia.org/wiki/Electrical_impedance" title="Electrical impedance">impedance</a> <img class="tex" alt="Z_R\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/26769a8d9f5658d17a0ecbd100a5011d.png"> of</p>
<dl>
<dd><img class="tex" alt="Z_R = R\,," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/d50b5595e875674c536ccfc0f566b3a8.png"></dd>
</dl>
<p>and a capacitor with <a href="http://en.wikipedia.org/wiki/Capacitance" title="Capacitance">capacitance</a> <img class="tex" alt="C\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/761e195b5e8196bddef5841e9451e2a6.png"> has impedance <img class="tex" alt="Z_C\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/7db97fb837f7fa2ac456e1928c54eb14.png"> of</p>
<dl>
<dd><img class="tex" alt="Z_C = \frac{1}{s C}\,," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/c3bf49dd3df5a0fa72e379ec9fe3332d.png"></dd>
</dl>
<p>where <img class="tex" alt="s = j \omega = \left(\sqrt{-1}\right) 2 \pi f\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/bc3c5cdb0ff8e689e7cf9064b87ef657.png"> and <img class="tex" alt="f\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/18f63800376271ee4b0efe1545744cd6.png"> is a <a href="http://en.wikipedia.org/wiki/Frequency" title="Frequency">frequency</a> of a pure <a href="http://en.wikipedia.org/wiki/Sine_wave" title="Sine wave">sine wave</a> input. That is, a capacitor's impedance is frequency dependent and a resistor's impedance is not.</p>
<h4><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Sallen%E2%80%93Key_topology&amp;action=edit&amp;section=4" title="Edit section: Example: Low-pass filter">edit</a>]</span> <span class="mw-headline" id="Example:_Low-pass_filter">Example: Low-pass filter<span id="Low-pass_configuration"></span></span></h4>
<p>An example of a unity-gain low-pass configuration is shown in Figure&nbsp;2.</p>
<div class="thumb tright">
<div class="thumbinner" style="width:402px;"><a href="http://en.wikipedia.org/wiki/File:Sallen-Key_Lowpass_General.svg" class="image"><img alt="" src="wikipedia-Sallen%E2%80%93Key_topology_pliki/400px-Sallen-Key_Lowpass_General.png" class="thumbimage" height="200" width="400"></a>
<div class="thumbcaption">Figure&nbsp;2: A unity-gain low-pass filter implemented with a Sallen–Key topology.</div>
</div>
</div>
<p>An <a href="http://en.wikipedia.org/wiki/Operational_amplifier" title="Operational amplifier">operational amplifier</a> is used as the buffer here, although an <a href="http://en.wikipedia.org/wiki/Emitter_follower" title="Emitter follower" class="mw-redirect">emitter follower</a> is also effective. This circuit is equivalent to the generic case above with</p>
<dl>
<dd><img class="tex" alt="
Z_1 = R_1, \quad Z_2 = R_2, \quad Z_3 = \frac{1}{s C_2}, \quad \text{and} \quad Z_4 = \frac{1}{s C_1}.\,
" src="wikipedia-Sallen%E2%80%93Key_topology_pliki/7300216719b4ced8ebfdf4555dc2f1ec.png"></dd>
</dl>
<p>The <a href="http://en.wikipedia.org/wiki/Transfer_function" title="Transfer function">transfer function</a> for this second-order unity-gain low-pass filter is</p>
<dl>
<dd><img class="tex" alt=" H(s) = \frac{ \overbrace{ ( 2 \pi f_c )^2 }^{ \omega_c^2 } }{ s^2 + \underbrace{ 2 \pi \frac{ f_c }{Q} }_{\frac{\omega_c}{Q} = 2 \zeta \omega_c }s + \underbrace{( 2 \pi f_c )^2}_{\omega_c^2} } " src="wikipedia-Sallen%E2%80%93Key_topology_pliki/1e5ac9616bd95e9deacc6ed549526d98.png"></dd>
</dl>
<p>where the <a href="http://en.wikipedia.org/wiki/Cutoff_frequency" title="Cutoff frequency">cutoff frequency</a> <img class="tex" alt="f_c\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/4fb5fe667ccf80ac91eb432a97841688.png"> and <a href="http://en.wikipedia.org/wiki/Q_factor" title="Q factor">Q factor</a> <img class="tex" alt="Q\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/703d7e7da69a957c727a4fa68f18cfe6.png"> (i.e., <a href="http://en.wikipedia.org/wiki/Damping_ratio" title="Damping ratio">damping ratio</a> <span class="texhtml">ζ</span>) are given by</p>
<dl>
<dd><img class="tex" alt=" f_c = \frac{1}{ 2 \pi \sqrt{R_1R_2C_1C_2} } " src="wikipedia-Sallen%E2%80%93Key_topology_pliki/508beb5ce6fc5fbbf033dfe5db1cf2fd.png"></dd>
</dl>
<p>and</p>
<dl>
<dd><img class="tex" alt=" 2 \zeta = \frac{1}{Q} = \frac{\sqrt{R_1R_2C_1C_2}}{C_1} \left( \frac{1}{R_1} + \frac{1}{R_2} \right).\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/81d4caf6eb6e2fa6ed238a65d6adb8ae.png"></dd>
</dl>
<p>So,</p>
<dl>
<dd><img class="tex" alt=" Q = \frac{ \sqrt{ R_1 R_2 C_1 C_2 } }{ C_2 \left( R_1 + R_2 \right) }
\qquad" src="wikipedia-Sallen%E2%80%93Key_topology_pliki/d4dbeb31e4646a2b9bcfc819c3feb892.png"></dd>
</dl>
<p>The <img class="tex" alt="Q\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/703d7e7da69a957c727a4fa68f18cfe6.png"> factor determines the height and width of the peak of the <a href="http://en.wikipedia.org/wiki/Frequency_response" title="Frequency response">frequency response</a> of the filter. As this parameter increases, the filter will tend to "ring" at a single <a href="http://en.wikipedia.org/wiki/Resonance" title="Resonance">resonant</a> <a href="http://en.wikipedia.org/wiki/Frequency" title="Frequency">frequency</a> near <img class="tex" alt="f_c\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/4fb5fe667ccf80ac91eb432a97841688.png"> (see "<a href="http://en.wikipedia.org/wiki/LC_filter" title="LC filter" class="mw-redirect">LC filter</a>" for a related discussion).</p>
<p>A <a href="http://en.wikipedia.org/wiki/Filter_design" title="Filter design">designer</a> must choose the <img class="tex" alt="Q\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/703d7e7da69a957c727a4fa68f18cfe6.png"> and <img class="tex" alt="f_c\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/4fb5fe667ccf80ac91eb432a97841688.png"> appropriate for his application. For example, a second-order <a href="http://en.wikipedia.org/wiki/Butterworth_filter" title="Butterworth filter">Butterworth filter</a>, which has maximally flat passband frequency response, has a <img class="tex" alt="Q\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/703d7e7da69a957c727a4fa68f18cfe6.png"> of <img class="tex" alt="1/\sqrt{2}\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/fedffcc81b4e62a96b325560f9949bc3.png">.
 Because there are two parameters and four unknowns, the design 
procedure typically fixes one resistor as a ratio of the other resistor 
and one capacitor as a ratio of the other capacitor. One possibility is 
to set the ratio between <img class="tex" alt="C_1\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/566395fdb9f8ee7ad63964694596894a.png"> and <img class="tex" alt="C_2\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/8eb5e2d4eee5009e251d364c001649f0.png"> as <img class="tex" alt="n\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/a957404c96e59f1746f97ab668c8e1f8.png"> and the ratio between <img class="tex" alt="R_1\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/b470348460d00246917364481612f047.png"> and <img class="tex" alt="R_2\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/459eab5223a515136857608da5fe7223.png"> as <img class="tex" alt="m\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/79dd9720ffa5bbe026e23afc9ab4df3c.png">. So,</p>
<dl>
<dd><img class="tex" alt="R_1=mR,\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/8781cef909698794b2bcff3d63711449.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="R_2=R,\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/424ee2fde05ba4ecba44b5c4b5dcbf89.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="C_1=nC,\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/28446a01942718b8be8c69eb3638b619.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="C_2=C.\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/a68f5741f619efdc2517bf934494b665.png"></dd>
</dl>
<p>Therefore, the <img class="tex" alt="f_c\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/4fb5fe667ccf80ac91eb432a97841688.png"> and <img class="tex" alt="Q\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/703d7e7da69a957c727a4fa68f18cfe6.png"> expressions are</p>
<dl>
<dd><img class="tex" alt=" f_c = \frac{1}{2\pi\ RC\sqrt{mn}},\, " src="wikipedia-Sallen%E2%80%93Key_topology_pliki/e02fd89a63f6e5d7d55899969c8961e8.png"></dd>
</dl>
<p>and</p>
<dl>
<dd><img class="tex" alt=" Q = \frac{\sqrt{mn}}{m+1}. " src="wikipedia-Sallen%E2%80%93Key_topology_pliki/439d0ac54fee63fb6068ef6f9ecbf613.png"></dd>
</dl>
<div class="thumb tright">
<div class="thumbinner" style="width:402px;"><a href="http://en.wikipedia.org/wiki/File:Sallen-Key_Lowpass_Example.svg" class="image"><img alt="" src="wikipedia-Sallen%E2%80%93Key_topology_pliki/400px-Sallen-Key_Lowpass_Example.png" class="thumbimage" height="200" width="400"></a>
<div class="thumbcaption">Figure&nbsp;3: A low-pass filter, which is implemented with a Sallen–Key topology, with <i>f</i><sub>c</sub>=15.9&nbsp;kHz and <i>Q</i>&nbsp;=&nbsp;0.5.</div>
</div>
</div>
<p>For example, the circuit in Figure&nbsp;3 has an <img class="tex" alt="f_c\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/4fb5fe667ccf80ac91eb432a97841688.png"> of <img class="tex" alt="15.9\,\text{kHz}\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/91a7a364c0a1ec037d9783e6b1e8ed9b.png"> and a <img class="tex" alt="Q\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/703d7e7da69a957c727a4fa68f18cfe6.png"> of <img class="tex" alt="0.5\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/54f7ef1990f1ea415ae1614c33249483.png">. The <a href="http://en.wikipedia.org/wiki/Transfer_function" title="Transfer function">transfer function</a> is given by</p>
<dl>
<dd><img class="tex" alt="H(s)=\frac{1}{1+\underbrace{C_2(R_1+R_2)}_{\frac{2 \zeta}{\omega_c} = \frac{1}{\omega_c Q} }s+\underbrace{C_1C_2R_1R_2}_{\frac{1}{\omega_c^2}}s^2}," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/8439245263ae583007485ccc03f440fe.png"></dd>
</dl>
<p>and, after substitution, this expression is equal to</p>
<dl>
<dd><img class="tex" alt="H(s)=\frac{1}{1+\underbrace{RC(m+1)}_{\frac{2 \zeta}{\omega_c} = \frac{1}{\omega_c Q} }s+\underbrace{mnR^2C^2}_{\frac{1}{\omega_c^2}}s^2}" src="wikipedia-Sallen%E2%80%93Key_topology_pliki/ac0ddefa6da1ce68f5b951cddefbafbc.png"></dd>
</dl>
<p>which shows how every <img class="tex" alt="(R,C)\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/7f2323f32be76f4461ed6ce364b4c97b.png"> combination comes with some <img class="tex" alt="(m,n)\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/66a077796cedb382e0f8a968de7336f6.png"> combination to provide the same <span class="texhtml"><i>f</i><sub><i>c</i></sub></span> and <span class="texhtml"><i>Q</i></span> for the low-pass filter. A similar design approach is used for the other filters below.</p>
<h4><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Sallen%E2%80%93Key_topology&amp;action=edit&amp;section=5" title="Edit section: Example: High-pass filter">edit</a>]</span> <span class="mw-headline" id="Example:_High-pass_filter">Example: High-pass filter</span></h4>
<div class="thumb tright">
<div class="thumbinner" style="width:402px;"><a href="http://en.wikipedia.org/wiki/File:Sallen-Key_Highpass_Example.svg" class="image"><img alt="" src="wikipedia-Sallen%E2%80%93Key_topology_pliki/400px-Sallen-Key_Highpass_Example.png" class="thumbimage" height="200" width="400"></a>
<div class="thumbcaption">Figure&nbsp;4: A specific Sallen–Key high-pass filter with <i>f</i><sub>c</sub>=72&nbsp;Hz and <i>Q</i>&nbsp;=&nbsp;0.5.</div>
</div>
</div>
<p>A second-order unity-gain high-pass filter with <img class="tex" alt="f_c\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/4fb5fe667ccf80ac91eb432a97841688.png"> of <img class="tex" alt="72\,\text{Hz}\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/8c5dd5b3c34be7c1454d29f470e286b7.png"> and <img class="tex" alt="Q\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/703d7e7da69a957c727a4fa68f18cfe6.png"> of <img class="tex" alt="0.5\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/54f7ef1990f1ea415ae1614c33249483.png"> is shown in Figure&nbsp;4.</p>
<p>A second-order unity-gain high-pass filter has the transfer function</p>
<dl>
<dd><img class="tex" alt=" H(s) = \frac{s^2}{s^2+\underbrace{2\pi(\frac{f_c}{Q})}_{2 \zeta \omega_c = \frac{\omega_c}{Q}}s+\underbrace{(2\pi f_c)^2}_{\omega_c^2}}, " src="wikipedia-Sallen%E2%80%93Key_topology_pliki/afb6e6da5a44ca1c4f5258c96d64bff9.png"></dd>
</dl>
<p>where cutoff frequency <img class="tex" alt="f_c\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/4fb5fe667ccf80ac91eb432a97841688.png"> and <img class="tex" alt="Q\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/703d7e7da69a957c727a4fa68f18cfe6.png"> factor are discussed above in the <a href="#Low-pass_configuration">low-pass filter</a> discussion. The circuit above implements this transfer function by the equations</p>
<dl>
<dd><img class="tex" alt=" f_c = \frac{1}{2\pi\sqrt{R_1R_2C_1C_2}}\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/b1f1e7408989dd86610b977781131260.png"></dd>
</dl>
<p>(as before), and</p>
<dl>
<dd><img class="tex" alt=" \frac{1}{2\zeta} = Q = \frac{\sqrt{R_1R_2C_1C_2}}{R_1(C_1+C_2)}.\," src="wikipedia-Sallen%E2%80%93Key_topology_pliki/7b4ae1c026a3778c3d15f7b9dea5041d.png"></dd>
</dl>
<p>So</p>
<dl>
<dd><img class="tex" alt=" 2 \zeta f_c = \frac{ f_c }{ Q } = \frac{ C_1 + C_2 }{ 2 \pi R_2 C_1 C_2  }. " src="wikipedia-Sallen%E2%80%93Key_topology_pliki/b1ab08f6d1123143350f2c1bcd03c4a3.png"></dd>
</dl>
<p>Follow an approach similar to the one used to design the low-pass filter above.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Sallen%E2%80%93Key_topology&amp;action=edit&amp;section=6" title="Edit section: VCVS Example: Bandpass configuration">edit</a>]</span> <span class="mw-headline" id="VCVS_Example:_Bandpass_configuration">VCVS Example: Bandpass configuration</span></h2>
<div class="thumb tright">
<div class="thumbinner" style="width:402px;"><a href="http://en.wikipedia.org/wiki/File:VCVS_Filter_Bandpass_General.svg" class="image"><img alt="" src="wikipedia-Sallen%E2%80%93Key_topology_pliki/400px-VCVS_Filter_Bandpass_General.png" class="thumbimage" height="270" width="400"></a>
<div class="thumbcaption">Figure&nbsp;5: A bandpass filter realized with a VCVS topology.</div>
</div>
</div>
<p>An example of a non-unity-gain bandpass filter implemented with a 
VCVS filter is shown in Figure&nbsp;5. Although it uses a different 
topology and an operational amplifier configured to provide 
non-unity-gain, it can be analyzed using similar methods as with the <a href="#Generic_Sallen.E2.80.93Key_topology">generic Sallen–Key topology</a>. Its transfer function is given by:</p>
<dl>
<dd><img class="tex" alt="H(s) = \frac{\overbrace{\left(1+\frac{R_b}{R_a}\right)}^{G} \frac{s}{R_1 C_1}}{s^2 +
  \underbrace{\left( \frac{1}{R_1 C_1} + \frac{1}{R_2 C_1} + \frac{1}{R_2 C_2} - \frac{R_b}{R_a R_f C_1} \right)}_{2 \zeta \omega_0 = \frac{\omega_0}{Q}} s +
  \underbrace{\frac{R_1 + R_f}{R_1 R_f R_2 C_1 C_2}}_{\omega_0^2 = (2\pi f_0)^2}}" src="wikipedia-Sallen%E2%80%93Key_topology_pliki/04f1cc2781774a0c2f06ca400daf3f85.png"></dd>
</dl>
<p>The <a href="http://en.wikipedia.org/wiki/Center_frequency" title="Center frequency">center frequency</a> <span class="texhtml"><i>f</i><sub>0</sub></span> (i.e., the frequency where the magnitude response has its <i>peak</i>) is given by:</p>
<dl>
<dd><img class="tex" alt=" f_0=\frac{1}{2\pi}\sqrt{\frac{R_f+R_1}{C_1C_2R_1R_2R_f}} " src="wikipedia-Sallen%E2%80%93Key_topology_pliki/6e92cd5381d0bd37dc561e9e0510c00a.png"></dd>
</dl>
<p>The voltage divider in the negative feedback loop controls the gain. The "inner gain" <span class="texhtml"><i>G</i></span> provided by the operational amplifier is given by</p>
<dl>
<dd><img class="tex" alt=" G=1+\frac{R_b}{R_a}" src="wikipedia-Sallen%E2%80%93Key_topology_pliki/28251f7f5f0ca6b04ca2c73a6c465043.png"></dd>
</dl>
<p>while the amplifier gain at the peak frequency is given by:</p>
<dl>
<dd><img class="tex" alt=" A=\frac{G}{3-G}" src="wikipedia-Sallen%E2%80%93Key_topology_pliki/070a56fce00b40978ece25f6b492de35.png"></dd>
</dl>
<p>It can be seen that <span class="texhtml"><i>G</i></span> must be kept below 3 or else the filter will oscillate. The filter is usually optimized by selecting <span class="texhtml"><i>R</i><sub>2</sub> = 2<i>R</i><sub>1</sub></span> and <span class="texhtml"><i>C</i><sub>1</sub> = <i>C</i><sub>2</sub></span>.</p>
<p>Caution, The Rb and Ra in the figure above have been swapped around, 
this causes a gain differant to expected when using this circuit design.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Sallen%E2%80%93Key_topology&amp;action=edit&amp;section=7" title="Edit section: See also">edit</a>]</span> <span class="mw-headline" id="See_also">See also</span></h2>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Filter_design" title="Filter design">Filter design</a></li>
<li><a href="http://en.wikipedia.org/wiki/Electronic_filter_topology" title="Electronic filter topology">Electronic filter topology</a></li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Sallen%E2%80%93Key_topology&amp;action=edit&amp;section=8" title="Edit section: External links">edit</a>]</span> <span class="mw-headline" id="External_links">External links</span></h2>
<ul>
<li><a href="http://focus.ti.com/lit/an/sloa024b/sloa024b.pdf" class="external text" rel="nofollow">Texas Instruments Application Report: Analysis of the Sallen–Key Architecture</a></li>
<li><a href="http://www.analog.com/en/amplifiers-and-comparators/products/dt-adisim-design-sim-tool/Filter_Wizard/resources/fca.html" class="external text" rel="nofollow">Analog Devices filter design applet</a>&nbsp;– A simple online tool for designing active filters using voltage-feedback op-amps.</li>
<li><a href="http://www-k.ext.ti.com/SRVS/CGI-BIN/WEBCGI.EXE/,/?St=147,E=0000000000002472277,K=2597,Sxi=1,Case=obj%2826717%29" class="external text" rel="nofollow">TI active filter design source FAQ</a></li>
<li><a href="http://focus.ti.com/lit/ml/sloa088/sloa088.pdf" class="external text" rel="nofollow">Op Amps for Everyone&nbsp;– Chapter 16</a></li>
<li><a href="http://postreh.com/vmichal/papers/frequency_filters_with_high_attenuation_Radio2009VMJS.pdf" class="external text" rel="nofollow">High frequency modification of Sallen-Key filter - improving the stopband attenuation floor</a></li>
<li><a href="http://www.changpuak.ch/electronics/calc_08.php" class="external text" rel="nofollow">Online Calculation Tool for Sallen–Key Low-pass/High-pass Filters</a></li>
<li><a href="http://www.tedpavlic.com/teaching/osu/ece327/lab7_proj/lab7_proj_procedure.pdf" class="external text" rel="nofollow">ECE 327: Procedures for Output Filtering Lab</a>&nbsp;– Section 3 ("Smoothing Low-Pass Filter") discusses active filtering with Sallen–Key Butterworth low-pass filter.</li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Sallen%E2%80%93Key_topology&amp;action=edit&amp;section=9" title="Edit section: References">edit</a>]</span> <span class="mw-headline" id="References">References</span></h2>
<div class="reflist" style="list-style-type: decimal;">
<ol class="references">
<li id="cite_note-EE315A_Notes-0">^ <a href="#cite_ref-EE315A_Notes_0-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-EE315A_Notes_0-1"><sup><i><b>b</b></i></sup></a> <a href="https://ccnet.stanford.edu/cgi-bin/course.cgi?cc=ee315a&amp;action=handout_download&amp;handout_id=ID126954294624704" class="external text" rel="nofollow">"EE315A Course Notes - Chapter 2"-B. Murmann</a></li>
<li id="cite_note-SallenKey-1"><b><a href="#cite_ref-SallenKey_1-0">^</a></b> <span class="citation Journal">Sallen, R. P.; E. L. Key (1955-03). "A Practical Method of Designing RC Active Filters". <i>IRE Transactions on Circuit Theory</i> <b>2</b> (1): 74–85.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.atitle=A+Practical+Method+of+Designing+RC+Active+Filters&amp;rft.jtitle=IRE+Transactions+on+Circuit+Theory&amp;rft.aulast=Sallen&amp;rft.aufirst=R.+P.&amp;rft.au=Sallen%2C%26%2332%3BR.+P.&amp;rft.date=1955-03&amp;rft.volume=2&amp;rft.issue=1&amp;rft.pages=74%E2%80%9385&amp;rfr_id=info:sid/en.wikipedia.org:Sallen%E2%80%93Key_topology"><span style="display: none;">&nbsp;</span></span></li>
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